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Category $\mathcal{O}$ of semisimple Lie algebra has been understood very well. One can decompose the category into different blocks by central characters, and evey block is Noetherian and Aritian, and with finite many irreducible objects.

I would like to know the corresponding structure of category $O$ of Kac moody algebra, especially for affine Lie algebra. For infinite dimensional Lie algebra, the central of universal envoloping algebra is big and complicated. For Affine Lie algebra, maybe one can decompose the category by different levels. Then let's fix a level $k$, and consider the corresponding category $O_k$. For convenience, we only consider representation with integral weight decomposition.

My question is

  1. For different noncritical $k$, can we decompose $\mathcal{O}_k$ into different blocks, and each of then contains only one integral irreducible representation.
  2. For critical $k$, it should be even more complicated, how to think of it?
  3. How about general Kac Moody algebra, is there any general theorem on this decomposition?
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    $\begingroup$ A good reference for the noncritical category O (for any symmetrizable Kac-Moody algebra) is arxiv.org/abs/math/0305378 by Peter Fiebig. For the critical level, things are much more complicated, see e.g. the long series of papers by Frenkel-Gaitsgory. $\endgroup$ Commented Apr 23, 2010 at 17:52
  • $\begingroup$ Fiebig's paper is definitely helpful and has been published (I think in the arXiv version): MR2205072 (2006k:17040) 17B67 Fiebig, Peter (D-FRBG), The combinatorics of category O over symmetrizable Kac-Moody algebras. Transform. Groups 11 (2006), no. 1, 29–49. Then try Frenkel-Gaitsgory. For blocks, note that already in the classical case, nonintegral weights require special arguments using Jantzen's integral Weyl subgroups. It's important to be precise about what you mean here by "block", since the term is sometimes used more loosely. $\endgroup$ Commented Apr 23, 2010 at 19:25
  • $\begingroup$ Dear Jim, I don't know the structure of central even at noncritical level, is there anything like Harich-chandra theorem for affine Lie algebra? The block in my mind, is a subcategory which corresponds to a central character. Maybe a correct integral block should only contain one irreducible integrable representation. $\endgroup$
    – JJH
    Commented Apr 23, 2010 at 20:55
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    $\begingroup$ @xiyu The short answer is that Harish-Chandra's classical description of the center of a universal enveloping algebra of a finite dimensional semisimple Lie algebra doesn't generalize well. And a good definition of "block" might avoid referring to special features like central characters. $\endgroup$ Commented Apr 24, 2010 at 12:35

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